Note that, the QMSEP objective function is convex, since \(-\log (\Phi (Ax))\)  is convex in \(x\)  and the sum of convex function is also convex, c.f. [1]}.
As shown in appendix REF  the objective function of the UBMSEP formulation is convex for \((d_{1,k}, d_{2,k}) >(0,0), k\in {1,\hdots ,K}\) . To ensure a convex objective, the original UBMSEP problem is rewritten as
\(&\displaystyle \min _{x_\text{r}} \ -\sum _{k=1}^K \log \left( \text{erf}\left( \left(h_{\text{R},\theta ,k}^{s^*} -h_{\text{I},\theta ,k}^{s^*}\right) x_{\text{r}} \right)+\text{erf}\left( \left(h_{\text{R},\theta ,k}^{s^*} +h_{\text{I},\theta ,k}^{s^*}\right) x_{\text{r} } \right)\right) \\&\text{s.t.} \ \ \ \ \begin{bmatrix} H_{\text{R},\theta }^{s^*} - H_{\text{I},\theta }^{s^*} \\ H_{\text{R},\theta }^{s^*} + H_{\text{I},\theta }^{s^*}\end{bmatrix} x_\text{r} \ge 0, \ \ \ \ {x}_{\text{r},2m-1}+j{x}_{\text{r},2m} \in \mathcal {X} \ \ \textrm {for } m=1,\ldots ,M\textrm {.}\) 
